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Can you advise some good books covering areas which are required for serious probability theory studies (e.g. measure theory, functional analysis)? Preferably this book should have some problem sets to work on. Thanks!

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Should be community wiki. – Jyotirmoy Bhattacharya Sep 26 '10 at 1:04
The answers should be cw as well. Or do they become automatically? – Rasmus Sep 26 '10 at 14:47
Apparently only answers that appeared after the question was made cw become cw automatically. – Rasmus Sep 26 '10 at 14:51

For a broad overview in one book at a graduate level, it is hard to beat Foundations of Modern Probability (Second edition) by Olav Kallenberg. His presentation is classroom tested, and many standard complicated proofs have been made very concise. I'm a big fan of this book!

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Pollard, A User's Guide to Measure Theoretic Probability

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A quite new and german one: Klenke, Wahrscheinlichkeitstheorie.

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Also available in english:… (see Simon's answer). – Rasmus Sep 28 '10 at 10:32

Take a look at "Measure Theory and Probability" by A.K. Basu. This book has lots of problem sets in them and contains some serious probabilty theory. It developed measure theory with probability as the main application. Also through-out the book there are examples which were really helpful.

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Good one! Thanks a lot! – Alexey Kalmykov Sep 25 '10 at 18:56

Dudley, Real Analysis and Probability

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Jacod and protter is great.

Probability Theory: A Comprehensive Course by Klenke is worth looking at too.

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